Regular order-preserving transformation semigroups

Abstract

A semigroup S is said to be regular if for each a E S, a = aba for some b E S. A partial transformation alpha on a set is said to be almost identical if x alpha is not equal to x for at most a finite number of elements x in the domain of alpha. Let X be a partially ordered set. Let PTOP(X), TOP(X), LOP(X), UOP(X), VOP(X) and WOP(X) denote the order-preserving partial transformation semigroup on X, the full order-preserving transformation semigroup on X, the order-preserving 1-1 partial transformation semigroup on X, the semigroup of all order-preserving almost identical partial transformations of X, the semigroup of all order-preserving almost identical transformations of X and the semigroup of all order-preserving almost identical 1-1 partial transformations of X, respectively. Let Z and R denote the set of integers and the set of real numbers, respectively. In this abstract, the partial order on any subset of R is the usual partial order on R. The main results of this research are as follows: Theorem 1. If X is a chain which is order-isomorphic to a subset of Z, then TOP(X) is regular. Theorem 2. For any interval X of R, TOP(X) is regular if and only if X is closed and bounded. Theorem 3. If X is a chain, then all of PTOP(X), IOP(X), UOP(X), VOP(X) and WOP(X) are regular. Theorem 4. Let X be a partially ordered set which is not a chain and let S be one of PTOP(X), IOP(X), UOP(X) and WOP(X). Then S is regular if and only if X is isolated. Theorem 5. If X is a partially ordered set containing (i) disjoint components C1 and C2 with /C1/>1 or a subposet of the forms... Theorem 6. Let X be a partially ordered set and M(X) and m(X) denote the set of all maximal elements of X and the set of all minimal elements of X, respectively. If (i) X = M(X) union m(X) and (ii) for x E m(X) and y E M(X), x>y, then TOP(X) is regular. Theorem 7. Let X be a partially ordered set. If X has a maximum element alpha and a minimum element b such that for all distinct x, y E X-{a, b}, x and y are not comparable, then TOP(X) is regular.